7 7 law of cosines
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Jim SmithJim SmithJCHSJCHSSection 7-7Section 7-7
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THE LAW OF COSINES ALLOWS US TO SOLVE THE TRIANGLETHE LAW OF COSINES ALLOWS US TO SOLVE THE TRIANGLE
WHEN WE CAN’T USE THE LAW OF SINES. SUPPOSE WE KNOWWHEN WE CAN’T USE THE LAW OF SINES. SUPPOSE WE KNOW
THE 3 SIDES OF A TRIANGLE BUT NONE OF THE ANGLES.THE 3 SIDES OF A TRIANGLE BUT NONE OF THE ANGLES.
THE LAW OF COSINES WILL HELP US FIND THE ANGLES.THE LAW OF COSINES WILL HELP US FIND THE ANGLES.
aa
bbcc
AA
CCBB
aa² = b² + c² = b² + c² - 2bc cos A² - 2bc cos Ab² = a² + c² - 2ac cos Bb² = a² + c² - 2ac cos Bc² = a² + b² - 2ab cos Cc² = a² + b² - 2ab cos C
NOTICE WHEN YOU START WITH SIDE NOTICE WHEN YOU START WITH SIDE aa YOU END WITH ANGLE YOU END WITH ANGLE AA
WHEN YOU START WITH SIDE WHEN YOU START WITH SIDE bb YOU END WITH ANGLE YOU END WITH ANGLE BB
WHEN YOU START WITH SIDE WHEN YOU START WITH SIDE cc YOU END WITH ANGLE YOU END WITH ANGLE CC
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BE CAREFUL WHEN YOU PUT THIS IN YOUR CALCULATORBE CAREFUL WHEN YOU PUT THIS IN YOUR CALCULATOR
a² = b² + c² - 2bc cos Aa² = b² + c² - 2bc cos A
a = a = (( b² + c² - 2bc cos A b² + c² - 2bc cos A
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2 SIDES AND THE INCLUDED ANGLE ( SAS )2 SIDES AND THE INCLUDED ANGLE ( SAS )AA
BBcc
6688
aa
FIND SIDE aFIND SIDE a
a² = b² + c² - 2bc cos Aa² = b² + c² - 2bc cos A
a²a² = 8² + 6² - 2 ( 8 ) ( 6 ) cos 50 = 8² + 6² - 2 ( 8 ) ( 6 ) cos 50
a = ( 8² + 6² - 2 ( 8 ) ( 6 ) cos 50a = ( 8² + 6² - 2 ( 8 ) ( 6 ) cos 50
a = 6.2a = 6.2
5050°°
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3 SIDES ( SSS )3 SIDES ( SSS )
AA
BB
CC1010
1515
2222
FIND ANGLE AFIND ANGLE A
a² = b² + c² - 2bc cos Aa² = b² + c² - 2bc cos A
1515² = 22² = 22² + 10² - 2 ( 22 ) ( 10 ) COS A² + 10² - 2 ( 22 ) ( 10 ) COS A
225 = 484 + 100 – 440 COS A225 = 484 + 100 – 440 COS A
225 = 584 – 440 COS A225 = 584 – 440 COS A
-359 = - 440 COS A359 = - 440 COS A
-359359 = COS A = COS A
-440-440
22ndnd COS COS ( ( -359 / -440-359 / -440
A = 35.3A = 35.3°°
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Golf On a 180-yard hole, a golfer hits the ball Golf On a 180-yard hole, a golfer hits the ball 175 yards, 3° off-line. How far is the ball from the cup? 175 yards, 3° off-line. How far is the ball from the cup? A.10.6 yd C.42.1 ydA.10.6 yd C.42.1 ydB.15.0 yd D.12.2 ydB.15.0 yd D.12.2 yd
180180
175175
33°°XX
SASSAS
aa² = b² = b² + c² - 2 bc cos 3² + c² - 2 bc cos 3
a² = 180² + 175² - 2( 180 ) ( 175 ) cos 3a² = 180² + 175² - 2( 180 ) ( 175 ) cos 3
a = 180² + 175² - 2( 180 ) ( 175 ) cos 3a = 180² + 175² - 2( 180 ) ( 175 ) cos 3
a = 10.55a = 10.55
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